Problem: Find $\dfrac{d}{dx}\left[e^{(6x-7x^2)}\right]$. Choose 1 answer: Choose 1 answer: (Choice A) A $(6-14x)e^{(6x-7x^2)}$ (Choice B) B $e^{(6x-7x^2-1)}$ (Choice C) C $e^{(6x-7x^2)}$ (Choice D) D $\ln(6-14x)e^{(6x-7x^2)}$
Solution: $e^{(6x-7x^2)}$ is an exponential function, but its argument isn't simply $x$. Therefore, it's a composite function. In other words, suppose $u(x)=6x-7x^2$, then $e^{(6x-7x^2)}=e^{u(x)}$. $\dfrac{d}{dx}e^{(6x-7x^2)}$ can be found using the following identity: $\dfrac{d}{dx}\left[e^{u(x)}\right]=e^{u(x)}\cdot u'(x)$ [Why is this identity true?] Let's differentiate! $\begin{aligned} &\phantom{=}\dfrac{d}{dx}e^{(6x-7x^2)} \\\\ &=\dfrac{d}{dx}e^{u(x)}&&\gray{\text{Let }u(x)=6x-7x^2} \\\\ &=e^{u(x)}\cdot u'(x) \\\\ &=e^{(6x-7x^2)}\cdot (6-14x)&&\gray{\text{Substitute }u(x)\text{ back}} \\\\ &=(6-14x) e^{(6x-7x^2)} \end{aligned}$ In conclusion, $\dfrac{d}{dx}e^{(6x-7x^2)}=(6-14x) e^{(6x-7x^2)}$